The aim of the unit is to give an introduction to algebraic topology with an emphasis on cell complexes, fundamental groups and homology.
Algebraic Topology concerns constructing and understanding topological spaces through algebraic, combinatorial and geometric techniques. In particular, groups are associated to spaces to reveal their essential structural features and to distinguish them. In cruder terms, it is about adjectives that capture and distinguish essential features of spaces.
The theory is powerful. We will give applications including proofs of The Fundamental Theory of Algebra and Brouwer's Fixed Point Theorem (which is important in economics).
Relation to other units
This is one of three Level M units which develop group theory in various directions. The others are Representation Theory and Galois Theory.
Students should absorb the idea of algebraic invariants to distinguish between complex objects, their geometric intuition should be sharpened, they should have a better appreciation of the interconnectivity of different fields of mathematics, and they should have a keener sense of the power and applicability of abstract theories.
The assimilation of abstract and novel ideas.
How to place intuitive ideas on a rigorous footing.
- Topological spaces – open sets, closed sets, product, quotient and subspace topologies, connectedness, path-connectedness, continuous maps, homeomorphisms, Hausdorff spaces
- The definition of the fundamental group and its calculation for the circle
- The Fundamental Theorem of Algebra
- Brouwer's Fixed Point Theorem
- Covering Spaces
- van Kampen's Theorem
- Graphs and free groups
- Singular homology
Reading and References
W. A. Sutherland, Introduction to metric and topological spaces, Clarendon Press, Oxford.
Munkres, Topology (2nd Edition), Pearson Education
Unit code: MATHM1200
Level of study: M/7
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Professor Jeremy Rickard
MATH20006 Metric Spaces and MATH33300 Group Theory
Methods of teaching
Lectures, problem sets and discussion of problems, student presentations.
Methods of Assessment
The pass mark for this unit is 50.
The final mark is calculated as follows:
- 80% from coursework (problem sets)
- 20% based on seminar presentations*
* Presentations graded on the understanding and insight they demonstrate and on the clarity, quality, lucidity and style of the delivery
For information resit arrangements, please see the re-sit page on the intranet.
Further exam information can be found on the Maths Intranet.